Problem: The lifespans of tigers in a particular zoo are normally distributed. The average tiger lives $20.5$ years; the standard deviation is $2.5$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a tiger living between $15.5$ and $28$ years.
$20.5$ $18$ $23$ $15.5$ $25.5$ $13$ $28$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $20.5$ years. We know the standard deviation is $2.5$ years, so one standard deviation below the mean is $18$ years and one standard deviation above the mean is $23$ years. Two standard deviations below the mean is $15.5$ years and two standard deviations above the mean is $25.5$ years. Three standard deviations below the mean is $13$ years and three standard deviations above the mean is $28$ years. We are interested in the probability of a tiger living between $15.5$ and $28$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the tigers will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the tigers will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of tigers between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular tiger living between $15.5$ and $28$ years is ${95\%} + \color{orange}{2.35\%}$, or $97.35\%$.